102 XIII INTEGRATION
and Supp(/£)c Uar For this purpose we observe that if K is the support of
/, then there exist finitely many indices af e I (1 ^ i :g ri) such that the Uaf
cover K. Hence ((3.18.2) and (12.6.4)) there exist n continuous mappings
ht'.X-* [0, 1] such that Supp(Af) is compact and is contained in Uai for
n
1 <£ i ^ 72, and such that £ Af(jc) = 1 for all x e K. Then the functions/; =/A,
»=i
satisfy the required conditions.
This already proves the uniqueness of f* : for by definition we must have
To prove the existence of a linear form ft on ,3rc(X) whose restriction to
jTc(Ua) is /ia for each a e I, it is enough to establish the following assertion:
given two finite sequences (ffi)i^i^m and (hj)i£jgn of functions belonging to
JTC(X), such that Supp(#f) c U~ for l^i^m and Supp^) c U^^ for
1 ^j ^n, and such that
for all x e Supp(/), then we have
1-1 /=!
Now, we have
and therefore
m m / n \
E^«X/^-)= Z T,H*t(f9ihj) •
i=J i=l\J=l /
Similarly,
But since Supp(/^f/i7.) is contained in UXi n Uftj, it follows from the hy-
potheses that ^ai(fyihj) = /*/»//<?!/*/), and our assertion follows.
It remains to be shown that the linear form \i so defined is a measure.
Let K be a compact subset of X, and define the Uai and the ht as at the be-
ginning of the proof. If Hf = Supp(Af), then by hypothesis there exists an write /in place of/| U when Supp(/) c U, and g in place of #u). The